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Chapter 16: Problem 9

Sketch the following polar rectangles. $$R=\\{(r, \theta): 1 \leq r \leq 4,-\pi / 4 \leq \theta \leq 2 \pi / 3\\}$$

Short Answer

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Question: Sketch the polar rectangle for the given set $$R=\\{(r, \theta): 1 \leq r \leq 4,-\pi / 4 \leq \theta \leq 2 \pi / 3\\}$$. Describe the process used to create the sketch. Answer: To sketch the polar rectangle, we first drew the polar coordinate axes with a circle and radial lines representing the angles. We marked the angles at $$-\pi / 4$$ and $$2 \pi / 3$$. Then, we drew two concentric boundary circles representing $$r = 1$$ and $$r = 4$$. We located the points where the radial lines intersected the inner and outer circles and labeled them A, B, C, and D. Finally, we connected the points to form the polar rectangle, which visually displays the region bounded by the given radius and angle limitations.

Step by step solution

01

Draw Polar Coordinate Axes

First, draw the polar coordinate axes on a plane, with a circle and radial lines representing the angles. Mark the angles $$-\pi / 4$$ and $$2 \pi / 3$$ by drawing radial lines at these angles.
02

Sketch Boundary Circles

Now, we will draw two circles to represent the bounds of radius $$r$$. The inner circle represents $$r = 1$$ and the outer circle represents $$r = 4$$. Make sure both circles are concentric with the polar coordinate origin.
03

Mark Angle Boundaries

Locate the points where the radial lines intersect the inner and outer circles, which represent the angle boundaries $$-\pi / 4$$ and $$2\pi / 3$$. Label these points as A, B, C, and D (Clockwise).
04

Connect Points to Form Rectangle

Connect the points A, B, C, and D to form the desired polar rectangle (A-B, B-C, C-D, and D-A). This rectangle represents the set $$R=\\{(r, \theta): 1 \leq r \leq 4,-\pi / 4 \leq \theta \leq 2 \pi / 3\\}$$ in polar coordinates. The sketched polar rectangle will visually display the region bounded by the given radius and angle limitations.

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